## Leonhard Euler

## Gabriel's Horn

Ezel Göktaş, Recai Efe Sunay

In this article, the underlying mathematical understanding of “Gabriel's Horn” is evaluated. According to some religious beliefs, Gabriel is supposed to blow his horn and announce Judgment Day when it comes. However, there is a notion that is unique and peculiar about the understanding of his horn when it is approached as a geometric figure. Furthermore, the uniqueness is amplified when it is evaluated on the Cartesian graph with the help of a very simple mathematical function, which is . In that case, the horn would have a finite volume but infinite surface area. This means the horn can be filled up with a finite amount of paint, but its surface cannot be painted. Now, how could a horn as such possibly exist?

This figure, in regard to Gabriel's Horn, is formed by taking the graph of with the domain x ≥ 1 and rotating it in three dimensions around the x-axis as shown below. The surface area and the volume of the horn between x = 1 and x = a where “a” approaches infinity is actually possible to calculate.

If the horn were to be split into many tiny circular segments, the area of a single circular segment could be calculated by the formula in correlation to the way that the area of a regular circle is calculated. In this case, r represents the radius of a single circular segment. Moreover, the radius is the distance between the line and the x-axis in the circular segment. Therefore, it could be stated that . Additionally, the horizontal width of the circular segment can be written as which is infinitely small. So now the volume of a single circular segment can be calculated as . To find the volume of the whole horn, the volume of all circular segments from x = 1 to infinity should be calculated and added up. To do that the integral should be set up to find the volume of the horn from x = 1 to x = a where a approaches to infinity.

Since a is assigned to be infinite, the expression is going to be equivalent to 0. Therefore, will be 1. So the result of this integral, is going to be equal to π. And that means the volume of the horn, where x is in the interval [1,∞), is actually π. Even though there isn’t a precise ending of the value π, the finite value of the volume could be understood as the value is between 3.14 and 3.15. As the paradox is also known as the “painter’s paradox,” it could be said that the horn could be filled with π amount of paint.

When it comes to the surface area of the horn, things get a little tougher. Even though the infinite value of the surface area was understood and invented through Cavalieri’s principles, nowadays the paradox could be understood through the use of calculus. The input values are from x = 1 to x = a, where a approaches to infinity in the graph. In correlation to the way the volume is calculated, the horn will be split into many tiny circular segments. The formula 2πr can be used to find the circumference of a single circular segment. But to compute the surface area of the whole horn, all of the circumferences of the segmented circular objects should be added up in the interval [1,∞). In order to carry out the calculation, the integral should be set up again as previously.

Then, the inner part of the root can be divided by and get . As it is known that is the derivative of the function , the whole equation can be written as

Moreover, as the way to find the derivative of is known, could be written and inserted in the equation above to obtain . Now, the integral could be set up that everything is in terms of x. As mentioned, the surface area would be calculated through the use of the formula 2πr. The analysis has previously shown that . The integral must be in the form where x starts from 1 and goes to infinity.

This integral is hard to evaluate, but it doesn’t have to be solved. It is known that the expression is always going to be greater than or equal to 1 since the domain for x is [1,∞), and it is known that the radius is always going to be greater than 0.

Although it’s hard to solve the integral that gives the surface area of the horn, the integral could be solved simply in the right which is known to be definitely smaller than the original integral with the given domain. When the integral on the right side is solved, the result is 2π ln(a) where a is supposedly infinity. As “a” approaches infinity, 2π ln(a) also approaches infinity.

Then, it could be said that the result of this integral is ∞. This means that the initial integral that gives the surface area is greater than an integral which has the solution ∞. If something is greater than infinity, then it must also be infinity. Finally, the understanding of the analysis can be concluded that Gabriel’s horn has an infinite surface area but finite volume.

References

Havil, Julian (2007). Nonplussed!: mathematical proof of implausible ideas. Princeton University Press. pp. 82–91.

Weisstein, Eric W. "Gabriel's Horn." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GabrielsHorn.html

Gabriel's Horn. Brilliant.org. May 27, 2021, https://brilliant.org/wiki/gabriels-horn/

Figure References

[1] “How to Find the Volume and Surface Area of Gabriel’s Horn” March 16, 2021, https://www.dummies.com/education/math/calculus/how-to-find-the-volume-and-surface-area-of-gabriels-horn

[2][4][5][6][7] Wikipedia. “Gabriel’s Horn.” May 27, 2021,https://en.wikipedia.org/wiki/Gabriel%27s_Horn

[3] “Parametric Functions” May 27, 2021, https://xaktly.com/ParametricEquations.html

Figure 1: Gabriel's Horn

Figure 2: The Volume of the Horn

However, can’t be used in this case because the circumference of the circular segments doesn’t change according to the x-axis. That means the rate of change is not horizontal. Since function is a curve, the rate of change can’t be horizontal ( ) or vertical ( ) as it was previously when the volume was calculated. It must be something else called , and could be defined by using the Pythagorean theorem. The distance between the two closest points on a curve is linear. So can be written. And then can be individually separated through the equation as .

Figure 3: The Volume of the Horn

Figure 4: The Surface Area of the Horn

Figure 5: An Inequality for the Surface Area

Figure 6: The Expression Going to Infinty

Figure 7: The Infinite Surface Area